Solve {ccc}{1}&{-8}&{3}{2}&{13}&{-1}{-1}&{8}&{-1}

$$\left| \begin{array} { c c c } { 1 } & { - 8 } & { 3 } \\ { 2 } & { 13 } & { - 1 } \\ { - 1 } & { 8 } & { - 1 } \end{array} \right|$$

Evaluate

$$58$$

Steps Using Sarrus' Rule

Find the determinant of the matrix using the method of diagonals.

$$det(\left(\begin{matrix}1&-8&3\\2&13&-1\\-1&8&-1\end{matrix}\right))$$

Extend the original matrix by repeating the first two columns as the fourth and fifth columns.

$$\left(\begin{matrix}1&-8&3&1&-8\\2&13&-1&2&13\\-1&8&-1&-1&8\end{matrix}\right)$$

Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.

$$13\left(-1\right)-8\left(-1\right)\left(-1\right)+3\times 2\times 8=27$$

Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.

$$-13\times 3+8\left(-1\right)-2\left(-8\right)=-31$$

Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.

$$27-\left(-31\right)$$

Subtract -31 from 27.

$$58$$

Steps Using Expansion by Minors

Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).

$$det(\left(\begin{matrix}1&-8&3\\2&13&-1\\-1&8&-1\end{matrix}\right))$$

To expand by minors, multiply each element of the first row by its minor, which is the determinant of the 2\times 2 matrix created by deleting the row and column containing that element, then multiply by the element's position sign.

$$det(\left(\begin{matrix}13&-1\\8&-1\end{matrix}\right))-\left(-8det(\left(\begin{matrix}2&-1\\-1&-1\end{matrix}\right))\right)+3det(\left(\begin{matrix}2&13\\-1&8\end{matrix}\right))$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.

$$13\left(-1\right)-8\left(-1\right)-\left(-8\left(2\left(-1\right)-\left(-\left(-1\right)\right)\right)\right)+3\left(2\times 8-\left(-13\right)\right)$$

Simplify.

$$-5-\left(-8\left(-3\right)\right)+3\times 29$$

Add the terms to obtain the final result.

$$58$$

Factor

$$2\times 29$$